INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL
J. Phys. A: Math. Gen. 37 (2004) 11547–11573
PII: S0305-4470(04)84279-3
A general stability criterion for droplets on structured
substrates
M Brinkmann1,2, J Kierfeld1 and R Lipowsky1
1
2
MPI für Kolloid und Grenzflächenforschung, D-14424 Potsdam, Germany
Interdisciplinary Research Institute, c/o IEMN, F-59652 Villeneuve d’Ascq, France
Received 2 August 2004, in final form 11 October 2004
Published 17 November 2004
Online at stacks.iop.org/JPhysA/37/11547
doi:10.1088/0305-4470/37/48/003
Abstract
Wetting morphologies on solid substrates, which may be chemically or
topographically structured, are studied theoretically by variation of the free
energy which contains contributions from the substrate surface, the fluid–
fluid interface and the three-phase contact line. The first variation of this free
energy leads to two equations—the classical Laplace equation and a generalized
contact line equation—which determine stationary wetting morphologies.
From the second variation of the free energy we derive a general spectral
stability criterion for stationary morphologies. In order to incorporate the
constraint that the displaced contact line must lie within the substrate surface,
we consider only normal interface displacements but introduce a variation of
the domains of parametrization.
PACS numbers: 02.30.Xx, 68.08.Bc, 47.20.Dr
List of symbols
α
β
δn
δc1
δs2
ε
θ
σ
ω
ψ
Vapour or low density phase
Wetting liquid or high density phase
nth variation
Variation including variation of arguments on contact line
Restricted second variation at a stationary shape
Small real number, subscript for varied quantities
Local contact angle
Rigid substrate
Torsion of a surface along its boundary
Normal displacement field
Line tension
0305-4470/04/4811547+27$30.00 © 2004 IOP Publishing Ltd Printed in the UK
11547
11548
µ
A = |A |
C
c1 , c2
c
c⊥
F
F̃
F
F
I
G
g̃
gij
hij
Lαβσ
L = |L |
L∗β
Lβ
αβ
αβσ
G
(s 1 , s 2 )
(t 1 , t 2 )
Sε Tε
D, G ⊂ R2
M
∇
∇2 = ∇ · ∇
N
n
R
r
t
V = |V |
U, V ⊂ R3
W
X
M Brinkmann et al
Eigenvalue to second variation of free energy
Surface tension
Area of surface A
Plane curve
Principal curvatures
Normal curvature, parallel to three-phase contact line
Normal curvature, perpendicular to three-phase contact line
Free energy, general
Free energy, pressure ensemble
Free energy of three-phase contact line
Interfacial free energy
Interval [0, Lαβσ ]
Gaussian curvature
Metric tensor or first fundamental form
Components of metric tensor
Components of extrinsic curvature tensor
Three-phase contact line
Length of space curve L
Length scale of line tension
Linear dimension of droplet
Arclength parameter
Width of αβ interface
Width of three-phase contact line
Capillary length
Coordinates on αβ interface
Coordinates on substrate σ
Coordinates of the varied three-phase contact line
Domain of parametrization
Mean curvature
Covariant operator
Laplace–Beltrami operator
Surface normal of αβ interface
Conormal of the αβ interface
Vector or parametrization of αβ interface
Vector or parametrization of three-phase contact line
Tangent of three-phase contact line
Volume of spatial region V
Neighbourhood
Wettability of substrate σ
Vector or parametrization of substrate σ
1. Introduction
The morphology of droplets wetting a solid substrate is strongly influenced by topographical
and/or chemical surface patterns [1]. During the past decade, much effort has been devoted,
both theoretically and experimentally, to this interplay between surface structure and wetting
morphologies. Experimental techniques such as micro-contact printing [2] or monolayer
lithography [3] allow the fabrication of imprinted or structured planar surfaces with tailored
patterns of lyophilic and lyophobic surface domains. The resulting wetting morphologies are
A general stability criterion for droplets on structured substrates
11549
governed by the patterns of different wettability over a wide range of droplet volumes, as has
been demonstrated in recent experiments on liquid channels [4, 5]. In these experiments, it
has also been shown that an increase of the droplet volume beyond a critical value leads to
a morphological transition from the liquid channel to a bulged configuration [4, 5]. Such a
transition offers the possibility to fuse different microcompartments of liquid in a controlled
manner, which is an attractive option for a variety of microfluidic applications [6].
The experimentally observed shapes of wetting droplets are theoretically obtained by
minimizing the free energy of the droplet. In general, this free energy contains contributions
from the substrate surface, from the fluid–fluid interface and from the line tension of the threephase contact line. For uniform substrate surfaces, this minimization leads to the classical
equations of Laplace and Young–Dupré. For chemically and topographically structured
substrate surfaces, a generalized contact line equation has been recently derived by Swain
and Lipowsky [7], see further below. These two equations provide necessary conditions for
the stationary configurations of the droplet but are not sufficient in order to determine whether
these configurations represent minima of the free energy and thus experimentally observable
shapes. Indeed, stationary droplet configurations may represent saddle points which exhibit
one or several unstable directions in the free energy landscape of configurations.
In order to determine the stability of a stationary droplet state, one has to study the
second variation of the droplet’s free energy. The issue becomes particularly relevant
at a morphological transition where two distinct shapes exchange their stability. Such
morphological transitions have been found for a variety of domain patterns consisting of
circular [8], striped [4, 9] and ring-shaped surface domains [10].
In the micrometre regime, the droplet shapes are governed by their interfacial free energies.
Contributions arising from the three-phase contact line are expected to become relevant at a
droplet size of about one hundred nanometres [1]. Thus, line tension effects are particularly
relevant for small wetting structures such as liquid bridges between wetting droplets [9]. In
the general framework of thermodynamics, a line contribution to the free energy of a wetting
droplet is defined as an excess free energy related to the structural perturbations along the
contact line [11]. Thermodynamic equilibrium implies that the interfacial tensions or free
energies are always positive. In contrast, the sign of the line tension may be positive or
negative [11]. Within effective interface models, the sign of the line tension is found to depend
on the details of the molecular interactions as reflected in the specific form of the interface
potentials, see, e.g., [12].
For uniform substrate surfaces, the equation of Young and Dupré represents the condition
of mechanical equilibrium of the three-phase contact line between liquid, vapour and substrate.
This equation has to be generalized for non-vanishing line tension as was first realized in [13]
and extended to certain surface domain geometries in [14, 15]. More recently, Swain and
Lipowsky [7] derived a rather general contact line equation which is valid for rigid substrates,
both topographically and chemically structured.
Several experimental studies have recently provided evidence for line tension effects on
wetting phenomena [16, 17]. In [17], the line tension corrections predicted by the contact
line equation in [7] have been determined by AFM measurements of the effective interface
potentials close to the contact line as well as contact angle measurements on droplets with
highly curved contact lines. Results derived from both methods agree and give a line tension
of the order of || ≈ 10−10 J m−1 . Depending on the liquid and the substrate, is observed to
have positive or negative sign [17]. In a recent experiment [18], the shape of nanometre-sized
polystyrene droplets has been measured using AFM. A systematic comparison between the
contact angles of droplets at different sizes shows that, in this case, the line tension attains a
value of || 0.5 ×10−10 J m−1 .
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M Brinkmann et al
In this paper, we start from basic concepts of interface thermodynamics using a general
expression for the free energy of a droplet on a structured substrate including interfacial and
line free energies. In the framework of differential geometry and variational calculus, we
derive the first variation of the free energy and recover the general contact line equation
for stationary states as previously derived by Swain and Lipowsky [7]. We then proceed to
calculate the second variation of the droplet’s free energy for such a stationary configuration
and, in this way, obtain a general criterion for the local mechanical stability of stationary
droplet shapes.
A major technical difficulty which has to be properly addressed in the variational
calculation is that the contact line of the deformed droplet shape must stay within the
substrate surface. In order to incorporate this constraint, previous approaches have used
shape deformations of the liquid–vapour interface which were parametrized both by normal
and by tangential displacements [19–21]. The tangential displacements can then be chosen in
such a way that the contact line stays within the substrate surface.
In this paper, we use a different calculational method for the derivation of the second
variation of free energy. The shape deformations are entirely parametrized in terms of normal
displacements of the initial configuration. Instead of introducing tangential displacements, the
necessary shift of the contact line is incorporated by varying the domain of parametrization of
the liquid–vapour interface and the wetted substrate. This scheme is rather transparent, and
one can easily see how the various terms of the second variation arise from the different terms
of the first variation.
The paper is organized as follows. A short summary of the thermodynamic description
of fluids wetting a chemically structured substrate is given in section 2. In section 3, we
consider general normal displacements of the droplet shape and calculate the variations of
general functionals defined on the droplet interface and contact line. The constraint that the
contact line should be displaced within the substrate surface is incorporated by introducing
variations of the domain of integration. In section 4, the variation of the droplet’s free energy
is derived in detail. The second variation leads to a general stability criterion as shown in
section 5.
2. Free energy
Let us consider a droplet of a liquid β with fixed volume wetting a rigid and inert substrate
σ . The wetted substrate is in contact with another third phase, denoted by α. The phase α
may be a vapour phase or another liquid phase. We assume that all phases are in thermal
and mechanical equilibrium. Concerning chemical equilibrium we may focus on two ideal
situations that can be realized in wetting experiments:
(i) The wetting phase β is non-volatile, that is, an exchange of molecules between α and β
is slow on the time scale of the experiment. Since all liquids are highly incompressible
at normal conditions, i.e., at temperatures and bulk pressures far from a critical point,
the volume Vβ of a β droplet is virtually constant throughout the wetting process. In
typical wetting experiments, the β phase may form disconnected droplets. Each β droplet
attains a local minimum of the droplet’s interfacial and line free energy for its given
individual volume. Due to condensation of molecules from the α phase the volume of a
β droplet might increase in time and, provided that mechanical equilibration is fast, we
may observe a sequence of local minima. If these conditions are fulfilled we will speak
of a volume ensemble because the droplet volume represents, in principle, an accessible
control parameter of the system.
A general stability criterion for droplets on structured substrates
11551
(ii) The exchange of molecules between the surrounding α phase and the β droplet is fast
and the system reaches chemical equilibrium. Whenever the number of particles in the
α phase is much larger than in the β phase we can regard the α phase as a reservoir of
molecules which fixes the chemical potential of each species in the entire system. In the
case of a one-component and simple liquid β coexisting with its vapour α we are able
to express the difference between the chemical potential µ, set by α, and the chemical
potential µ0 at coexistence of the bulk phases by the pressure difference Pα − Pβ across
the αβ interface. Thus, we refer to this situation as the pressure ensemble.
In general, the free energy F of the droplet consists of different contributions related
to the bulk of the droplet, its interfaces and the three-phase contact line [1]. Which of
these contributions govern its morphology depends on the length scale of the droplet. Also
gravitational and van der Waals forces can give relevant contributions to the free energy on
large and small scales, respectively.
2.1. Interfacial and contact line energies
At fixed intensive quantities the interfacial free energy F
is homogeneous in the area
Aij = |Aij | of each interface Aij between adjacent homogeneous phases i and j. Because of
chemical inhomogeneities of the substrate walls σ , both surface tensions ασ and βσ are
functions of the position X on the substrate surface Aσ . Thus, we cast the droplet’s interfacial
free energy into a form
F
= αβ Aαβ +
dA [
βσ (X) − ασ (X)],
(1)
Aβσ
where Aβσ is the surface of σ wetted by β.
To proceed with basic definitions, the three-phase contact line Lαβσ of the droplet is
defined as the set of points where the αβ interface terminates on the surface of σ , i.e.
Lαβσ = Aαβ ∩ Aσ . On a chemically homogeneous substrate wall the related free energy
F is simply proportional to the length Lαβσ = |Lαβσ | of the three-phase contact line.
However, on heterogeneous substrate walls one might expect a spatially non-uniform line
tension (X) and a line integral
F =
dL (X)
(2)
Lαβσ
accounts for the contact line free energy. One may also consider droplets wetting plane and
homogeneous crystalline or nano-patterned surfaces where the line tension is expected to
become a function of the local orientation of the contact line.
The line free energy F becomes comparable with the interfacial free energy if the linear
dimension, Lβ , of the droplet is of the order of the characteristic length [1]
(3)
L∗β = /
αβ 2αβσ αβ
as follows from dimensional analysis where αβ is the width of the αβ interface and αβσ is
the width of the contact line.
As long as the linear dimension of the droplet satisfies Lβ L∗β , the line free energy is
small compared to the interfacial free energies. This does not imply, however, that the line
tension has no observable consequences in this case. In fact, most estimates of the line tension
are based on contact angle measurements in the regime Lβ L∗β since the contact angle is
affected by the tension, see equation (64) below.
The regime Lβ < L∗β may not be accessible to experimental studies if the width of the
contact line is relatively small. The contact line is the intersection of three interfaces which
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M Brinkmann et al
will, in general, differ in their width. Usually, the width of the αβ interface is comparable to
the size of the molecules in the two fluid phases (exceptions are provided by phase coexistence
near a critical point, where the α and β phases become indistinguishable, and by the adsorption
of surfactant molecules which increase the effective interfacial width). If the substrate surface
is smooth on the molecular scale, the width of the contact line should also be governed by
the width of the αβ interface. In this case, the characteristic length L∗β is of the order of the
molecular size as well, and droplets with linear dimensions Lβ < L∗β are not accessible.
On the other hand, the magnitude of the line tension may be increased by the effect
of long-range van der Waals forces. As shown in appendix E, the corresponding effective
interface model leads, in the small gradient approximation, to the line tension contribution
vdW ≈
1 AH
6π θ lpin
(4)
from van der Waals forces. This contribution depends on the contact angle θ , the Hamaker
constant AH and the wetting layer thickness l = lpin where the interface potential has a point
of inflection. This line tension contribution is large for small contact angle θ , large Hamaker
constant AH and small thickness lpin . A large contribution vdW increases the characteristic
size L∗β ∼ , and the regime Lβ < L∗β may then become accessible.
2.2. Bulk energies
The droplet’s body Vβ or, to be precise, the space occupied by the liquid β, with volume
Vβ = |Vβ | is bounded by the surface Aαβ to the vapour α, and by the surface Aβσ to
the rigid substrate σ , and thus, is entirely determined by the shape and position of the αβ
interface. An analogue statement holds for the three-phase contact line Lαβσ and Aβσ , the
surface of σ wetted by β. Hence, the droplets free energy is a functional of the αβ interface
that is mathematically described by Aαβ . In order to find locally or globally stable droplet
configurations in contact to the substrate walls we have to minimize the free energy
F = F
+ F
(5)
under the subsidiary constraint Vβ = V allowing for a droplet volume that is equal to the
value of V . The volume ensemble can be seen as analogous to a canonical ensemble since the
number of particles in the incompressible phase β is fixed.
Provided that, in contrast to the volume ensemble, the pressure difference P = Pα − Pβ
between the fluid phases α and β is prescribed while Vβ is not fixed, i.e, the pressure ensemble
applies to the system, an additional term P Vβ in the free energy
F̃ = F
+ F + P Vβ
(6)
accounts for the work done or received during an exchange of volume with an external reservoir.
In the volume ensemble, however, the pressure difference P is a Lagrange multiplier which
has to be chosen such that the constraint of a constant volume is fulfilled. At fixed volume,
the last term in (6) is a constant and does not contribute to the free energy3 .
In the grand canonical ensemble, the pressure difference is fixed by the supersaturation
µ = µ − µ0 with µ being the actual, fixed value of the chemical potential in the container
and µ0 being the chemical potential at coexistence of bulk phases α and β. A quantitative
3
Note that P is the pressure difference between vapour α and liquid phase β which is set by the external reservoir
and identical to the pressure difference across the droplet’s αβ interface if and only if the droplet is in mechanical
equilibrium.
A general stability criterion for droplets on structured substrates
11553
relation between the pressure difference P and the shift µ in the chemical potential at
coexistence is given by the equation
µ =
P
+ O[(P )2 ],
ρα0 − ρβ0
(7)
with particle number densities ρi0 of phases i ∈ {α, β} at bulk coexistence.
Gravity does not affect the shape of droplets which are significantly smaller than the
capillary length
2
αβ
,
(8)
G =
a(ρβ − ρα )
where relevant contributions to the droplet free energy come from the interfaces and, for
droplets in the submicrometre regime, from the three-phase contact lines between adjacent
bulk phases [1]. Effects of gravity can be eliminated on a macroscopic length scale performing
wetting experiments with two liquids of nearly the same mass densities ρα and ρβ as well as
in space crafts or drop towers where the acceleration a is typically reduced by a factor of 10−4
to 10−6 compared to normal conditions on earth. The capillary length G is typically of the
order of millimetres or even larger.
3. Droplet shape
Small deformations of a liquid droplet β that is freely suspended in a fluid α can be represented
by normal displacements of the αβ interface with respect to the initial droplet configuration.
However, normal displacements are not sufficient to describe deformations of a droplet β
wetting the wall of a substrate σ . The shift of the contact line under general shape variations
may have a tangential component with respect to the αβ interface.
One obvious way to approach this problem is to introduce an additional displacement
along a tangential direction close to the boundary of the αβ interface as first introduced in
[19]. The tangential shift together with the normal displacement allows for the constraint of a
contact line gliding on the substrate walls. Within such an approach, it is hard to eliminate the
tangential displacement field in the final results. It seems that this problem has been overcome
in [20, 21] where displacement fields for each order of the variational problem have been
introduced including a tangential shift. Due to the constraint of a gliding contact line these
fields are not independent and subject to a condition that is derived in a long calculation in [21].
This condition can be employed to eliminate tangential components in the final expressions
for free energy variations.
In this paper, we use a different approach to treat the constraint of the contact line gliding
on the walls of the substrate σ . By constructing an extension of the αβ interface beyond the
surface of σ and by use of special coordinate systems close to the contact line we were able
to vary the domains of parametrizations of the αβ interface and of the surface of σ that is
wetted by β, respectively. This approach turns out to be computationally advantageous as it
naturally leads to the occurrence of boundary terms in the variations of physical quantities,
like interfacial energies or droplet volume. As we will show in the following, it enables
us to split up the variations into terms which are entirely related to either the variation of
the αβ interface or the variation of the contact line. This justifies a classification of these
variations into ‘surface’ and ‘boundary’ terms which will be marked with superscripts (s) and
(b), respectively.
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M Brinkmann et al
3.1. Parametrizations
It is reasonable to assume that the substrate wall σ , the droplet’s αβ interface and three-phase
contact line are smooth, that is, they exhibit no edges or kinks. The αβ interface of the droplet,
we denote by Aαβ , is parametrized by smooth functions R = (R 1 , R 2 , R 3 ) with
x = R(s 1 , s 2 )
(9)
in local coordinates (s 1 , s 2 ) ∈ D ⊂ R2 that are mapped onto Euclidean vectors x =
(x 1 , x 2 , x 3 ) ∈ Aαβ ⊂ R3 . A tangential plane to a given point of Aαβ is spanned by a
pair of linearly independent tangent vectors Ri = ∂R/∂s i for any regular parametrization.
√
By convention, the local normal vector N = (R1 × R2 )/ g on Aαβ points from the√liquid
√
β to the vapour α. In the definition above we used the local surface element g = det g
of the metric tensor g, or first fundamental form, with components gij = Ri · Rj . Within the
formalism of co- and contravariant components, elements of the matrix inverse g −1 of g will
be written as (g −1 )ij = g ij in local coordinates. The surface area of Aαβ is measured by an
integral
√
Aαβ =
dA =
d2 s g,
(10)
Aαβ
D
which is invariant under reparametrizations of the αβ interface.
The surface Aβσ , i.e., the part of the surface of σ wetted by the liquid β, is parametrized
by smooth functions X = (X1 , X2 , X3 ) with
x = X(t 1 , t 2 )
(11)
in local coordinates (t , t ) ∈ G ⊂ R and with a three-dimensional Euclidean vector
x ∈ Aβσ ⊂ R3 . Trivially, an inclusion Aβσ ⊂ Aσ holds with the surface of σ here denoted by
Aσ . Tangent vectors Xi = ∂X/∂t i define the metric tensor components gij∗ = Xi · Xj and a
√
√
√
local surface element g ∗ = det g ∗ . A corresponding normal vector M = (X1 × X2 )/ g ∗
on Aβσ points from the substrate σ to the liquid β. We will use the convention that all
quantities or operators referring to Aβσ or Aσ are marked with a star to distinguish them from
those defined on Aαβ .
The three-phase contact line Lαβσ = Aαβ ∩ Aσ , i.e, the set of points where the liquid β,
the vapour α and the substrate σ meet can be described as the rim of either the surface Aαβ or
Aβσ . Smooth functions r = (r 1 , r 2 , r 3 ) represent the contact line as a space curve
1
2
x = r()
2
(12)
with ∈ I ⊂ R and x ∈ Lαβσ ⊂ R3 . By reparametrization one may choose the parameter such that || is identical to the arclength on the contact line. Hence, we have I = [0, Lαβσ ]
with Lαβσ , the length of the contact line.
Apart from the local tangent t = ṙ/|ṙ| to Lαβσ we will use a conormal vector n = t × Nc
to the surface Aαβ which forms a right-handed local frame (n, t, Nc ) of R3 in every point of
Lαβσ . Here and below, we denote the total derivative with respect to the parameter by a dot.
The vector Nc is the local surface normal N of Aαβ restricted to points of Lαβσ . Provided that
the boundary curve r() has a positive orientation with respect to the surface normal N the
conormal n points outwards when seen from Aαβ . A conormal m = t × Mc to the surface Aβσ
yields a second orthonormal and right-handed frame (m, t, Mc ) of R3 with Mc , the surface
normal M of Aβσ restricted to points of Lαβσ . The local contact angle θ between Aαβ and
Aβσ is defined by a scalar product
cos θ = n · m = Nc · Mc ,
(13)
A general stability criterion for droplets on structured substrates
11555
(α)
N
Mc
c
θ
m
t
n
(β)
(σ)
Figure 1. Cut through a droplet β on a substrate σ , perpendicular to the contact line Lαβσ . The
droplet β is surrounded by a vapour phase α. In points on Lαβσ we denote the surface normal of the
αβ interface Aαβ by Nc and the surface normal to the substrate Aσ by Mc . The outward-pointing
normal n is perpendicular to t and Nc yielding a local orthonormal local frame of R3 . The local
frame (n, t, Nc ) on Aαβ is tilted by the local contact angle θ against the corresponding local frame
(m, t, Mc ) on Aσ .
as illustrated in figure 1. The relative orientation of Aαβ and Aβσ with respect to the
parametrization of Lαβσ implies that n = m and Nc = Mc provided θ = 0 holds.
By the divergence theorem one can rewrite the droplet volume as a surface integral over
Aαβ and Aβσ . We have
1
1
Vβ =
dV =
dA R · N −
dA X · M
3 Aαβ
3 Aβσ
Vβ
1
1
√
=
d2 s g R · N −
d2 t g ∗ X · M,
(14)
3 D
3 G
which turns out to be a computationally advantageous starting point when varying the αβ
interface. The signs in front of the first and the second term are due to the different orientations
of the normal N to Aαβ and the normal M to Aβσ when seen from the bulk of β.
3.2. Extending surfaces
We are able to construct a small stripe Sαβ along the contact line Lαβσ that provides a smooth
continuation A¯αβ = Aαβ ∪ Sαβ of the αβ interface beyond the contact line, see also appendix
A. Here, the term ‘smooth’ means in particular that the normal curvature is defined in any
point and along any direction in the corresponding local tangential plane on the extended
surface A¯αβ . A continuation A¯βσ = Aβσ ∪ Sβσ of the wetted surface of σ can be easily
constructed since Aβσ is embedded in the smooth substrate wall Aσ . Both continuations are
necessary to account for variations in the position of the contact line under small deformations
of the droplet. It is convenient to introduce special parametrizations x = R(s 1 , s 2 ) on A¯αβ
and x = X(t 1 , t 2 ) on A¯σβ in the vicinity of Lαβσ , for details see appendix A. Given a
parametrization of Lαβσ as a space curve x = r() we have
r() = R(S(), ) = X(T (), ),
(15)
with s = S() = 0, t = T () = 0 and s = t = for all ∈ I . As the second coordinate
is now given by the parameter we may introduce the simplifying notation s = s 1 and t = t 1
that is clear as long as we refer to this special coordinate system.
1
1
2
2
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M Brinkmann et al
Both coordinate systems arise naturally in the construction of the extensions that we
outline in appendix A. The absolute value |s| is the arclength on a coordinate line defined by
= const between a particular point x = R(s, ) and the contact line. The point x will be
located on the extending stripe Sαβ if the coordinate s satisfies the inequality 0 s < S̄(),
and x will lie on Aαβ if s < 0 holds. The rim of the extending stripe is parametrized in these
coordinates by a function s = S̄(). Analogue statements hold for the extending stripe Sβσ
and coordinates (t, ) of A¯βσ . Here, the rim of the extending stripe is given by a function
t = T̄ (l) in local coordinates. Now, for the domains of parametrization D ⊂ R2 and G ⊂ R2
with
Aαβ = R(D)
Aβσ = X(G ),
we can find extended domains D̄ ⊂ R2 and G¯ ⊂ R2 such that
A¯αβ = R(D̄)
A¯βσ = X(G¯).
(16)
(17)
Expressed in the special coordinates the domains parametrizing the extending stripes Sαβ and
Sβσ assume the particular form
D̄\D = [0, S̄] × I
G¯\G = [0, T̄ ] × I .
(18)
As shown appendix A the local tangent t to the contact line satisfies
t = R2 |s=0 = X2 |t=0 ,
(19)
i.e, is identical to the second local tangent vector on the αβ interface and of the surface of σ
wetted by β, respectively, in points on the contact line. The remaining tangent vectors
R1 |s=0 = n
X1 |t=0 = m
(20)
in points on the contact line represent the local conormal n to the αβ surface and the local
conormal m to the wetted surface of σ , respectively.
3.3. Variation of shape
To test for stationarity or local stability of the αβ interface we have to impose small
deformations to the αβ interface before calculating changes in the interfacial free energy.
Given a parametrization R of the extended αβ interface A¯αβ we represent the varied, extended
αβ interface (A¯αβ )ε by
x = Rε (s 1 , s 2 ) = R(s 1 , s 2 ) + δR(s 1 , s 2 ),
(21)
3
1 2
¯ )ε ⊂ R . We choose a displacement δR = εψN along
with (s , s ) ∈ D̄ and x ∈ (Aαβ
the surface normal N of the unvaried surface A¯αβ where ψ(s 1 , s 2 ) is a smooth function of
coordinates s i on the extended surface A¯αβ and ε is a small and real number. Now, in
order to describe the physical part of the varied αβ interface (Aαβ )ε ⊂ (A¯αβ )ε , the domain
of parametrization D is varied to Dε ⊂ D̄ with (Aαβ )ε = R(Dε ) ⊂ (A¯αβ )ε . The latter
inclusions presuppose that |ε| is sufficiently small. This procedure can be seen as cutting off
the redundant part of the varied extended αβ interface that is lying in the bulk of the substrate
σ . The remaining physically relevant part (Aαβ )ε of the αβ interface is bounded by the varied
contact line (Lαβσ )ε that is defined by the relation (Lαβσ )ε = (Aαβ )ε ∩ Aσ . Under the shift of
the contact line, the surface of σ that is wetted by β has to be varied to (Aβσ )ε = X(Gε ) ⊂ A¯βσ
with a corresponding varied domain Gε ⊂ G¯.
It is one of the central issues of this paper to calculate the variation of the domains D and
G of parametrization from this geometrical constraint. Equation (15) serves as a starting point
to determine the coordinates of the varied contact line. This condition can be rewritten as
rε () = Rε (Sε (), ) = X(Tε (), )
(22)
A general stability criterion for droplets on structured substrates
11557
in coordinates s = Sε () and t = Tε () of the varied contact line (Lαβσ )ε on the varied
extended αβ interface (A¯αβ )ε and the extended wetted surface A¯βσ of σ , respectively. The
varied domains of parametrization Dε and Gε are expressed in the form
D̄\Dε = [Sε , S̄] × I
G¯\Gε = [Tε , T̄ ] × I
(23)
using the special coordinate systems. Differentiation of equation (22) with respect to the small
parameter ε yields with equation (20) the relation
(24)
δ 1 r = nδ 1 S + Nc ψ = mδ 1 T .
d
1
The notation δ (·) = dε (·) ε=0 is an abbreviation for the first variation of a function under the
deformation of the αβ interface. By subsequent scalar multiplication of equation (24) with
n, Nc and m, and by comparison with the definition of the local contact angle θ defined in (13)
we find
ψ
δ 1 S = ψ cot θ
δ1T =
δ 1 r = mδ 1 T
,
(25)
sin θ
where all quantities refer to the unvaried contact line. The requirement of smooth curvatures
on the contact line, as guaranteed by continuation of the αβ interface described in the preceding
section and appendix A, ensures that the variations (25) are uniquely defined.
3.4. Surface and boundary terms
As a first application of formula (25) we will calculate the variation of a general class of
functionals
√
E {Aαβ } =
dA E =
d2 s gE,
(26)
Aαβ
D
that depend on the configuration Aαβ of the αβ interface. The geometrical quantity E defined
on the extended surface A¯αβ is a function of local coordinates (s 1 , s 2 ) ⊂ D̄. Note that the first
and the second variation of the functional Eε = E {(Aαβ )ε } which depends on the real number
ε through the varied αβ interface (Aαβ )ε are defined by
dEε d2 Eε 1
2
δ E=
.
(27)
δ E {ψ} =
dε dε2 ε=0
ε=0
In particular, this definition implies that the iterated first variation is identical to the second
variation δ 1 (δ 1 E ) = δ 2 E .
To continue, we calculate the first variation of E as the first derivative
d
√
1
2 √
δ E {ψ} =
d s gε Eε =
d2 s δ 1 ( g)E
dε
D
D
ε=0
ε
d
√
2
1
2 √
2 √
d s gδ E +
d s gE −
d s gE .
(28)
+
dε
D
Dε
D
ε=0
The last term in the general expression (28) comes from the variation of the domain of
parametrization (23) which is due to the shift of the contact line. In the special coordinate
system defined in section 3.2 and using first variations as given in (25) we may expand the
first coordinate of the contact line as Sε = εψ cot θ + O(ε2 ) around ε = 0. By construction of
√
the coordinate system we have g = 1 + O(s) around s = 0, as follows from (19) and (20).
The last term of (27) leads to a ‘boundary’ term
d
√
√
d2 s gE −
d2 s gE δ 1 E b {ψ} =
dε
Dε
D
ε=0
Sε
d
√
=
d
ds gE =
d E cot θ ψ.
(29)
dε
I
0
ε=0
I
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M Brinkmann et al
to the first variation of the functional E . The remaining terms in (28) give rise to a ‘surface’
term
√
1 s
2
1√
δ E {ψ} =
d s δ ( g)E +
d2 s gδ 1 E.
(30)
D
D
Note that this surface term would be zero if the surface Aαβ itself were not varied. We conclude
that an equivalent surface term in the first variation of an analogous class of functionals
E ∗ {Aαβ } =
dA E ∗ =
d2 t g ∗ E ∗ ,
(31)
Aβσ
G
which are defined as an integral over the surface of σ wetted by β, is absent, i.e., we have
δ 1 E ∗ s {ψ} = 0.
(32)
Here, we use an expansion Tε = εψ/ sin θ + O(ε ) of the first coordinate of the varied contact
line around ε = 0 to show that the boundary term to the first variation of the functional (31)
is given by
d
1 ∗b
2
∗
2
∗ ∗
∗
ds g E −
d t g E δ E {ψ} =
dε
Gε
G
ε=0
Tε d
Eψ
.
(33)
=
d
dt g ∗ E ∗ =
d
dε
sin
θ
I
0
I
ε=0
√
An expansion g ∗ = 1 + O(t) can be performed around t = 0, as follows again from relations
(19) and (20).
To proceed with the computation of the surface term (30), we calculate the first variation
√
δ 1 ( g) of the local surface element using the Gauss–Weingarten map which assumes the form
2
j
∂i N = hi Rj
(34)
in local coordinates. We find local tangent vectors to the varied αβ interface
j
(Rε )i = Ri + Rj hi εψ + Nε∂i ψ,
(35)
j
hi
of the extrinsic curvature tensor are positive on a
where, in our definition, all elements
spherical surface. From equation (35) we derive the expression
(gε )ij = gij + 2hij εψ + O(ε2 )
(36)
for the varied components (gε )ij of the metric tensor gij , and we are able to compute the first
√
variation of the local surface element g. Given a non-singular matrix a and an arbitrary
matrix b, the useful identity
d
det(a + εb)|ε=0 = Tr(a −1 b) det(a)
(37)
dε
allows us to calculate a directional derivative of a determinant. By use of formula (37) and
the expansion (36) we obtain
√
√
√
δ 1 ( g) = ghii ψ = 2 gMψ
(38)
i
as the first variation of the local surface element. The trace M = hi 2 is the mean curvature
and equals the mean value of the two principal curvatures. Since the trace of the extrinsic
curvature tensor is invariant under any reparametrization which conserves the orientation of
the surface, the mean curvature represents a local geometrical invariant of a surface [22].
Inserting equation (38) into the surface term (30) and with the boundary term (29) we
arrive at a general expression
dA 2MEψ +
dA δ 1 E +
dL E cot θ ψ
(39)
δ 1 E {ψ} =
Aαβ
Aαβ
Lαβσ
A general stability criterion for droplets on structured substrates
11559
for the first variation of the functional E which itself is a functional of the normal displacement
field ψ on A¯αβ . The first variation of the analogous functional E ∗ {Aαβ } of the wetted surface
of σ , however, is given by a boundary term
E∗ψ
.
(40)
dL
δ 1 E ∗ {ψ} =
sin θ
Lαβσ
Finally, we compute the variation of a general class of functionals B which are defined
as an integral over the contact line. Since the substrate σ is rigid and thus its surface is not
varied the contact line is entirely determined by the shape and position of the αβ interface.
Dropping, for the moment, the assumption that is an arclength parameter on the contact line
we may write
B{Aαβ } =
dLB =
d|ṙ|B,
(41)
Lαβσ
I
where B is a function of local geometrical quantities. These quantities, for instance, depend
on the position on the varied αβ interface and on the surface of σ and, hence, change under a
shift of the contact line. The variation of the functional B leads to two terms
d|ṙ|δc1 B +
d δc1 (|ṙ|)B,
(42)
δ 1 B{ψ} =
I
I
where the first term arises from the variation of the geometrical quantity that is due to the
variation of the αβ interface and the accompanying shift of the contact line. The subscript
c indicates a total variation under the shift of the contact line including all arguments. The
second term describes the effect of local lengthening or shortening of the contact line under
the normal displacement by ψ. The first variation of the line element becomes
ṙ · δ 1 ṙ
= t · (ṁδ 1 T + mδ 1 Ṫ ),
(43)
|ṙ|
where we used the identity (24) and the Leibnitz rule. With the definition of the geodesic
curvature
ṫ · m
t · ṁ
cg∗ =
=−
(44)
|ṙ|
|ṙ|
of the contact line with respect to the surface of σ and the orthogonality relation m · t = 0 we
can rewrite expression (43) as
cg∗ ψ
δc1 (|ṙ|) = t · ṁδ 1 T = |ṙ|
.
(45)
sin θ
The first variation of the functional B is then given by
cg∗ Bψ
1
1
,
(46)
δ B{ψ} =
dL δc B +
sin θ
Lαβσ
δc1 (|ṙ|) =
where we still have to compute the total variation δc1 B of the integrand B while allowing for
the tangential shift of the contact line.
First, we consider the situation where the integrand B depends on the position of the
contact line on the varied extended αβ interface and the surface of σ , i.e., on the functions
Sε () and Tε (). Since, by definition, the second coordinate is not varied, we have
dBε (Sε , Tε , ) δc1 B =
= δ 1 B + (∂s B)δ 1 S + (∂t B)δ 1 T
(47)
dε
ε=0
for the total first variation of the integrand B due to the variational shift of the contact line.
The first variation of the coordinates of the contact line, δ 1 S and δ 1 T , is given in (25).
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M Brinkmann et al
In the other situation, the integrand B does not depend on the position of the contact
line but on the direction of the contact line on the substrate wall, i.e., on the unit vector m.
This situation is of interest when considering homogeneous but anisotropic substrates σ . For
simplicity, we assume that the anisotropic surface of σ is planar so that we can describe the
orientation by a single angle φ between the conormal m and a fixed unit vector e lying in the
plane of the surface, i.e., cos φ = m · e. This leads to a first variation
δc1 (cos φ) = −sin φδc1 φ = e · δc1 m.
(48)
The first variation of the conormal is calculated in appendix C with the result δc1 m = −tδ 1 Ṫ
for a planar substrate, see equation (C.16). By the simple relation sin φ = −t · e we conclude
that δc1 φ = −δ 1 Ṫ . We are now able to calculate the first variation of the quantity B under
the rotation of the contact line induced by the shift. Instead of relation (47) being valid for a
spatially dependent B we now have
δc1 B =
dBε (φε ) = δ 1 B − (∂φ B)δ 1 Ṫ
dε ε=0
(49)
for the total first variation of an integrand B due to the rotation of the contact line under the
variation.
4. Variation of free energy
Using the results for the variation of generic functionals derived in the preceding section we
will now calculate first and second variations of the interfacial free energy and line energy
for droplets wetting a rigid substrate wall. The result for the second variation is valid only for
stationary droplet shapes but this restriction may be of no significance since applied to test for
local stability.
In the pressure ensemble, a stationary configuration of the αβ interface is reached
whenever the first variation of the total free energy vanishes under any small deformation. In
contrast, for non-volatile and incompressible droplets the volume ensemble holds and thus a
constraint of a constant volume has to be imposed on the class of configurational variations
of the αβ interface. In principle, the conditions of stationarity do not reveal any information
about the local or global stability of the droplet.
The volume term in the free energy (6) can be seen, depending on the ensemble, either
as a Langrangian term or as an energy term that accounts for the exchange of volume with
an external reservoir. This observation implicates that the set of stationary configurations
with fixed volume and the set of stationary configurations at fixed Laplace pressure are
identical. But because of the constraint of constant volume, the set of locally mechanically
stable configurations in the pressure ensemble is a subset of all locally mechanically stable
configurations in the volume ensemble. In what follows, we will focus our attention mostly
on the volume ensemble.
As opposed to local stability, global stability can only be inferred if all branches of
stationary configurations in the wetting geometry are known. Wetting geometries which allow
such an exhaustive analytical solution of the first variational problem are rare and, in many
cases, exhibit non-analytical solutions as well. In a numerical investigation, this has been
demonstrated by Lenz and Lipowsky for droplets wetting a planar lyophobic substrate which
is decorated with a lyophilic ring [10].
A general stability criterion for droplets on structured substrates
11561
4.1. First variation
A general expression of the first variation including both interfacial free energy and line
free energy has been derived by Swain and Lipowsky [7]. The condition for a mechanical
equilibrium of the αβ interface is still given by the equation of Laplace while the equation
of Young–Dupré has to be modified due to line tension. Effects of line tension are strong
whenever the curvature of the contact line is large, as it is the case for small droplets, or if
the line tension depends strongly on the position on the substrate. Already some decades
ago, Gretz considered the equilibrium shape of small spherical droplets on a chemically
homogeneous substrate including line tension [13]. He derived a quantitative expression for
the deviation of the local contact angle at mechanical equilibrium of a small droplet from a
macroscopic droplet. Our calculation, however, allows us to consider the first variation of
arbitrarily shaped droplets.
As introduced in section 2 we write the free energy of small droplets in the form
F̃ = F
+ F + P Vβ ,
(50)
where the influence of gravity has been neglected. We assume, for the moment, that the droplet
exchanges volume with an external reservoir at a given pressure difference P = Pα − Pβ
between the fluid phase α and the wetting liquid β. If we do not allow changes in the droplet
volume Vβ the last term in the free energy (50) becomes a constant. In this case P can be
seen as a Lagrange multiplier, see section 2.
According to equation (1) we split the interfacial free energy F
into the free energies Fαβ
and Fβσ , which are related to the αβ interface and the surface of σ wetted by β, respectively.
The variation of the interfacial free energy Fαβ = αβ Aαβ can be performed with the general
expression (39) for the particular case E = 1 which leads to surface and boundary contributions
s
b
{ψ} + δ 1 Fαβ
{ψ}
δ 1 Fαβ {ψ} = δ 1 Fαβ
dA 2Mψ +
= αβ
Aαβ
dL cot θ ψ .
(51)
Lαβσ
On the other hand, the interfacial free energy which stems from the wetted part of σ is given
by an integral
dAW,
(52)
Fβσ = −
αβ
Aβσ
where the wettability W = (
ασ − βσ )/
αβ ranges between −1 and 1. As the variation
of the αβ interface leaves the shape of the rigid substrate σ unchanged we find no surface
s
= 0, see (32). Therefore, the only contribution
contribution to the first variation, i.e., δ 1 Fβσ
to the first variation of the interfacial free energy Fβσ is due to the boundary and obtained by
setting E ∗ = W in equation (33) as
Wψ
b
.
(53)
{ψ} = −
dL
δ 1 Fβσ {ψ} = δ 1 Fβσ
sin
θ
Lαβσ
Another contribution to the total free energy of the droplet comes from line tension
and, thus, is sensitive to the variation of the three-phase contact line only. According to
equation (46) the first variation of the line energy (2) consists of two terms
cg∗ ψ
.
(54)
dL δc1 +
dL
δ 1 F {ψ} =
sin θ
Lαβσ
Lαβσ
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M Brinkmann et al
The first term in (54) describes variations of the line energy which emerge from a positional
change of the contact line and the chemical inhomogeneity on the surface of σ . Applying
equation (47) we find
δc1 = (∂t )δ 1 T =
ψm · (∇ ∗ )
,
sin θ
(55)
where we used the covariant derivative ∇ ∗ = Xi ∂i defined on the extended surface A¯βσ . The
first variation of the line energy is then given by
ψ
1
(cg∗ + m · ∇ ∗ ).
δ F {ψ} =
dL
(56)
sin θ
Lαβσ
The last term of the free energy F̃ in the volume ensemble contains the volume Vβ of
the liquid droplet. By definition the pressure difference P is fixed by the external volume
reservoir and, thus, is not varied. A variation of the droplet volume can be easily performed
in the representation by surface integrals (14) with the result
1
1
ψ
1
1
1
dA(N · δ R + δ N · R) +
dL r · (cos θ Nc − Mc )
δ Vβ {ψ} =
,
(57)
3 Aαβ
3 Lαβσ
sin θ
where we applied the identities (15), (39) and (40). The varied αβ interface is represented
through normal displacements, and following equation (21) we have δ 1 R = ψN. The first
variation of the normal N to the αβ interface is given by the simple expression δ 1 N = −∇ψ,
as shown in appendix B, see equation (B.5). Here, ∇ = Ri ∂i is the covariant derivative on the
extended surface A¯αβ . Then, by use of Gauss theorem
−
dA R · ∇ψ =
dA ψ∇ · R −
dL n · rψ
(58)
Aαβ
Aαβ
Lαβσ
on Aαβ and employing the identity ∇ · R = R · Ri = = 2 we arrive at the first variation
1
ψ
1
.
(59)
dA ψ +
dL r · (Nc cos θ − n sin θ − Mc )
δ Vβ {ψ} =
3 Lαβσ
sin θ
Aαβ
i
δii
The vector decomposition Mc = Nc cos θ − n sin θ (compare figure 1) on the contact line
shows that the boundary term of the first variation is zero. Hence, we find a first variation of
the droplet volume that is simply given by
δ 1 Vβ = δ 1 Vβs =
dA ψ,
(60)
Aαβ
and has no boundary term.
Collecting the terms from the first variations of the free energies of the αβ interface (51),
the wetted surface of σ (53), the line energy (56) and the droplet volume (60) we finally arrive
at the expression
δ 1 F̃ {ψ} = δ 1 F
{ψ} + δ 1 F {ψ} + P δ 1 Vβ {ψ}
dA(2
αβ M + P )ψ
=
Aαβ
+
dL
Lαβσ
ψ
[
αβ (cos θ − W ) + cg∗ + m · ∇ ∗ ]
sin θ
(61)
as the first variation of the droplet free energy in the pressure ensemble which consists of
a surface and a line integral. Since the normal displacement field ψ on the αβ interface is
arbitrary, we can infer from the condition δ 1 F̃ {ψ} = 0 of mechanical equilibrium that the
A general stability criterion for droplets on structured substrates
11563
arguments of both the surface integral and the line integral in (61) have to vanish identically.
This leads to the equation of Laplace
2
αβ M = −P = Pβ − Pα
on
Aαβ ,
(62)
and the generalized equation of Young and Dupré
αβ cos θ = ασ − βσ − cg∗ − m · ∇ ∗ on
Lαβσ
(63)
for stationary droplet shapes. The latter result was obtained first in its general form by Swain
and Lipowsky [7]. Here, the local equilibrium contact angle θ is not only a function of the
position, but also depends on the orientation m and geodesic curvature cg∗ of the contact line
with respect to the surface of the substrate.
In the following, we will briefly discuss the somewhat different situation of a homogeneous
and planar, but anisotropic surface of a crystalline or nanopatterned substrate σ . The conormal
m and a fixed direction e on the surface of σ form an angle φ which was already introduced
in (3.4). A variation of the line energy now leads to
1
(64)
dL[cg∗ δ 1 T − (∂φ )δ 1 Ṫ ],
δ F =
Lαβσ
where the dot denotes a derivative with respect to an arclength parameter on the contact
line. Note that on a planar surface of σ we find a relation cg∗ = φ̇. By a partial integration
with respect to the arclength parameter for a closed (or periodic) contact line on the planar
surface we arrive at the first variation
dL cg∗ + ∂φ2 δ 1 T ,
(65)
δ 1 F =
Lαβσ
and the following generalization of the equation of Young and Dupré:
αβ cos θ = ασ − βσ − cg∗ + ∂φ2 on Lαβσ .
(66)
It may happen on a surface of a crystalline or periodically nanopatterned substrate that the
second derivative ∂φ2 becomes large and positive for certain contact line orientations. Since
the cosine of the contact angle ranges between −1 and 1 the geodesic curvature cg∗ of the contact
line with respect to the surface of σ has to go to small positive values in a small range of
orientations which leads to a strong faceting of the droplet’s three-phase contact line. Such
faceting effects have recently been observed for liquid alloy droplets wetting a Si surface [23].
4.2. Second variation
The second variation of the free energy can be obtained by iterated variation of the result
(61) for the first variation. Although the calculation is in principle straightforward using the
techniques we have provided for treating boundary terms arising from the contact line, a much
simpler expression for the second variation can be obtained if we restrict ourselves to the
calculation of the second variation for stationary droplet shapes for which the first variation
has to vanish. Consequently, the conditions of Laplace (62) and Young–Dupré (63) have to be
fulfilled. In the following, we will write for the ‘restricted second variation’, i.e., the second
variation evaluated at a stationary shape, the symbol δs2 (.) to distinguish from the ordinary
operator δ 2 (.) of second variation. Then, we obtain from iterated variation of the first variation
(61) at such a stationary shape the expression
dA ψδ 1 (2
αβ M + P )
δs2 F̃ {ψ} =
Aαβ
+
dL
Lαβσ
ψ 1
δ [
αβ (cos θ − W ) + c∗ + m · ∇ ∗ ].
sin θ c
(67)
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M Brinkmann et al
The final result for the restricted second variation has a much simpler form compared to
the full second variation because the arguments in the surface and in the line integral of (61)
are zero in any point of the αβ interface and contact line, respectively, for a stationary droplet
shape. Hence, variations of the domain of integration and the variation of the local surface or
line element need not be considered in (67). Using the result (B.12) for the first variation of
the mean curvature we find
2 s
dA 2ψδ 1 M
δs F̃
{ψ} = αβ
Aαβ
= αβ
Aαβ
dA ψ[−∇ 2 + (2G − 4M 2 )]ψ
(68)
as the surface contribution to the restricted second variation of the free energy in the pressure
ensemble. The second-order differential operator ∇ 2 = ∇ · ∇ is the Laplace–Beltrami
operator on A¯αβ .
The second term in (67) gives the boundary contributions to the restricted second variation
and leads to four terms: the first term describes the variation of the local contact angle θ
under the deformation of the αβ interface; a second term arises from the position-dependent
wettability W on the substrate and the shift of the contact line; the third and the fourth term
are related to the change of the geodesic curvature and the position-dependent line tension.
Expressed in an orthonormal coordinate system, as given by the special coordinate systems
close to the contact line introduced in section 3.2, the co- and the contravariant components of
any tensor become identical and the components of the curvature tensor can be identified with
local normal curvatures or torsions of the surface in the directions of the coordinate lines.
In the following, we will denote the components of the extrinsic curvature tensor on the
contact line by special symbols. In particular, we use c⊥ = h11 for the normal curvature of
the αβ interface perpendicular to the contact line, ω = h12 = h21 for the torsion or winding of
the αβ interface along the contact line, and c = h22 for the normal curvature parallel to the
∗
, ω∗ and c∗ are introduced with respect to the special
contact line. Corresponding symbols c⊥
orthonormal coordinate system on the surface of σ .
Employing the general result for the first total variation (47) of a quantity defined on the
contact line and from the definition of the local contact angle (13) we find
cos θ Mc · ∂s N + Nc · ∂t M
ψ,
(69)
δc1 (cos θ ) = Mc · δ 1 N +
sin θ
where the first variation δ 1 N and the derivatives ∂s N and ∂t M are taken on the contact line.
By use of the Gauss–Weingarten equation (34) together with (19) and (20), the first variation
of the surface normal (B.5), and some trigonometry the above expression simplifies to
∗
− c⊥ cos θ )ψ.
δc1 (cos θ ) = sin θ n · ∇ψ + (c⊥
(70)
A further contribution to the second term in the restricted second variation of the free energy
(67) arises from a position-dependent wettability W of the substrate wall. The first variation
under the shift of the contact line is simply given by
ψm · ∇ ∗ W
.
(71)
sin θ
Whenever we deal with large droplets of linear dimension Lβ L∗β , line energies become
irrelevant, and we can neglect all terms which contain the line tension . In this case, the
boundary contribution to the restricted second variation of the free energy reduces to
dL ψ(
αβ n · ∇ + u
)ψ,
(72)
δs2 F
b {ψ} =
δc1 W = (∂t W )δ 1 T =
Lαβσ
A general stability criterion for droplets on structured substrates
coming from the interfacial energies where u
denotes the expression
∗
c⊥
m · ∇∗ W
− c⊥ cot θ −
,
u
= αβ
sin θ
sin2 θ
11565
(73)
∗
defined on the contact line and depending on normal curvatures c⊥ and c⊥
as well as the
derivative of the wettability W in the direction of the conormal m.
If we consider the local stability of small droplets with linear dimension in the range
Lβ L∗β we have to include the remaining last two terms in (67), both related to line energy,
into our consideration. Iterated variation leads to the restricted second variation
ψ 1
δ (m · ∇ ∗ + cg∗ ).
δs2 F {ψ} =
dL
(74)
sin θ c
Lαβσ
The first term can be further transformed into
δc1 (m · ∇ ∗ ) = ∇ ∗ · δc1 m + m · (∇ ∗ ⊗ ∇ ∗ ) · mδ 1 T ,
(75)
containing the first variation of the conormal with respect to the surface of σ . The second
term leads to two new terms
δc1 (cg∗ ) = δc1 cg∗ + cg∗ m · (∇ ∗ )δ 1 T ,
(76)
which account for the variation of geodesic curvature cg∗ and the coupling of the geodesic
curvature to the position-dependent line tension . The lengthy calculations of the first
variation of the geodesic curvature cg∗ and the first variation of the conormal m with respect to
the substrate are presented in appendix C with the result
(77)
δc1 cg∗ = −δ 1 T̈ − cg∗ 2 + G∗ δ 1 T
and
δc1 m = −t δ 1 Ṫ − Mc cg∗ δ 1 T .
The first variation of the two bracketed terms in expression (74) becomes
˙ 1 Ṫ
δc1 (m · ∇ ∗ + cg∗ ) = −δ 1 T̈ − δ
∗
+ cg m · ∇ ∗ + m · (∇ ∗ ⊗ ∇ ∗ ) · m − cg∗ 2 + G∗ δ 1 T .
(78)
(79)
Recalling that the first factor in (74) is simply given by ψ/ sin θ = δ 1 T we may perform a
partial integration in the parameter along the closed contact line which yields a symmetric
representation
2
dL[(δ 1 Ṫ )2 + v(δ 1 T )2 ]
(80)
δs F {ψ} =
Lαβσ
of the restricted second variation of the line energy where v is an abbreviation for
v = cg∗ m · ∇ ∗ + m · (∇ ∗ ⊗ ∇ ∗ ) · m − cg∗ 2 + G∗ .
(81)
In a similar manner, the surface term of the second variation (68) assumes a symmetric form
after an integration by parts which eliminates the term ψn · ∇ψ under the line integral in (72).
Including both the interfacial and the line energy contributions, the restricted second
variation of the free energy of a droplet wetting a rigid substrate σ can be cast into the final
form
dA[(∇ψ) · (∇ψ) + (2G − 4M 2 )ψ 2 ]
δs2 F̃ {ψ} = αβ
Aαβ
dL
+
Lαβσ
(ψ̇ − θ̇ ψ cot θ )2 + (v + u
sin2 θ )ψ 2
,
sin2 θ
(82)
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M Brinkmann et al
where the functions u
and v , defined by (73) and (81), respectively, depend on the normal
∗
as well as on the geodesic curvature cg∗ of the contact line and the
curvatures c⊥ and c⊥
∗
Gaussian curvature G with respect to the surface of σ , and derivatives of the wettability W
and line tension in the direction of the conormal m.
The restricted second variation (82) of the droplet’s free energy is the main result of
this paper. It agrees with the result given in [20] with details of its derivation in [21]. As
expected, expression (82) is symmetric in the displacement field ψ on the αβ interface. Note
also that we have eliminated the pressure difference P in equation (82) as the restricted
second variation is taken at a stationary configuration and as we have implicitly made use of
the Laplace equation (62).
In the case of a uniform line tension, the function (81) depends only on the line tension
and on the intrinsic curvature of the substrate wall and contact line, that is, the Gaussian
curvature G∗ and the geodesic curvature cg∗ . Introducing ripples to the boundary of the αβ
interface lengthens the contact line as indicated by the first term in the boundary integral (82).
This term has the same sign as the line energy.
5. Spectral stability criterion
The second variation of the free energy of a locally stable droplet has to be non-zero and
positive for any deformation of the αβ interface that is compatible with the constraint of a
constant droplet volume. Hence, positivity of the restricted second variation δs2 F̃ {ψ} implies
local stability of the droplet. The final, symmetric form (82) shows that the restricted second
variation of the free energy is a quadratic form in the displacement field ψ and positivity
can be inferred from its spectrum of eigenvalues µ on the subspace of volume-conserving
displacements ψ. The smallest eigenvalue yields the desired information about local stability.
A droplet configuration is locally stable if and only if the smallest eigenvalue is positive.
The spectrum of eigenvalues can be found by diagonalizing the quadratic form δs2 F̃ {ψ} on
the subspace of volume-conserving displacements ψ using the well-known extremal property
of eigenfunctions, which establishes the equivalence to an eigenvalue problem for a partial
differential equation with specific boundary conditions [24].
For non-volatile and incompressible liquids, the droplet volume has to be conserved
under any deformation of the αβ interface. For the stability analysis of configurations that
are stationary under such a constraint we have to test the restricted second variation (82) for
positivity on a subspace of displacements that enforces the volume-constraint up to linear
order in ψ as shown in appendix D.
The expression of the first variation of the droplet volume (57) shows that this condition
is equivalent to
dA ψ = 0,
(83)
δ 1 Vβ {ψ} =
Aαβ
which defines a linear subspace U⊥ of all displacement fields on the αβ interface. The
displacement field is normed by
dA ψ 2 = 1,
(84)
Aαβ
and all fields ψ that satisfy (84) are said to belong to the unit sphere S of elongation fields on
the αβ interface.
Now, the quadratic form δs2 F̃ {ψ} can be diagonalized by extremizing on the restricted
set S of normalized displacement fields ψ on the αβ interface [24]. The associated Lagrange
A general stability criterion for droplets on structured substrates
11567
multiplier for the restriction to S is the eigenvalue µ. For non-volatile liquids with conserved
volume, further restriction of S to the subset S⊥ = S ∩ U ⊥ of fields has to be considered in
the stability analysis instead of S . Equivalently, we can introduce another Lagrange multiplier
ν, and find that the quadratic form δs2 F̃ {ψ} is diagonalized by displacement fields ψ which
are local extrema of the functional Q{ψ} = δs2 F {ψ} − µ
Aαβ
dA ψ 2 − ν
dA ψ,
(85)
Aαβ
where, so far, no restrictions are made upon ψ. Both parameters µ and ν have to be chosen
such that the subsidiary conditions of normalization (84) and of conserved volume (83) are
satisfied for an extremal field ψ. The extremizing displacement field ψ is an eigenfunction in
the subspace of volume-conserving displacements ψ, and the parameter µ the corresponding
eigenvalue of the second variation δs2 F {ψ} restricted to U⊥ . We finally conclude after a partial
integration of the symmetric form Q{ψ} that the extremal condition δ 1 Q{ψ} = 0 is equivalent
to the inhomogeneous version of a partial differential eigenvalue equation
−∇ 2 ψ + (2G − 4M 2 )ψ = µψ + ν
on
Aαβ .
(86)
The partial integration of Q{ψ} further shows that the displacement field ψ has to fulfil a
linear boundary condition
v
1
+ u
ψ + r{ψ} = 0 on Lαβσ ,
n · ∇ψ +
(87)
αβ sin2 θ
where the function r{ψ} on the contact line has a form
d d ψ
1
.
(88)
r{ψ} = −
αβ sin θ d d sin θ
If the smallest eigenvalue µ to a solution ψ of the partial differential equation (86) with
boundary condition (87) is positive, the restricted second variation (82) is also positive, which
implies local stability of the stationary configuration.
A uniform local contact angle θ and line tension in (88) yields the simplified expression
r{ψ} = −ψ̈/(
αβ sin2 θ ), which applies to simple droplet morphologies on plane substrates
with uniform wettability and line tension, as we will consider in a following publication.
6. Conclusion
In this work, we have derived the most general form of the second variation (82) of the
free energy for a stationary droplet wetting a rigid substrate with spatially heterogeneous
surface topography, wettability and line tension. A new variational technique was introduced
to treat the motion of the contact line under arbitrary deformations of the droplet’s liquid–
vapour interface. Within this variational technique it is sufficient to consider only normal
displacements of the liquid–vapour interface. Using this formalism, we are able to recover
expression (61) for the first variation of the free energy which has been obtained previously
by Swain and Lipowsky [7] and leads to a generalized equation of Young and Dupré (63) for
heterogeneous substrates including line tension effects. We have also considered the situation
of a homogeneous but anisotropic line tension for a planar substrate. In this case, we obtain
expression (65) for the first variation of the free energy and a different generalization (66) of
the equation of Young and Dupré which can give rise to a faceting of wetting morphologies.
Starting from the result (82) for the restricted second variation we derived the partial
differential equation (86) with boundary conditions (87) which allows a systematic stability
analysis of stationary droplet shapes and will be applied to a simple wetting geometry in a
following publication.
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M Brinkmann et al
t
Figure 2. Illustration of the construction of the extending stripes Sαβ and Sβσ . As outlined in
appendix A we construct an array of planes P perpendicular to the tangent t of Lαβσ . Provided
that the contact line Lαβσ is smooth the planes P do not intersect each other in a sufficiently small
neighbourhood U(x) around a point x ∈ Lαβσ .
Our variational scheme may be useful in a variety of other contexts. Possible applications
include the stability analysis of fluid membrane vesicles adsorbed onto a substrate with spatially
heterogeneous chemical composition. Due to the presence of additional bending energies, an
expression for the second variation of the elastic free energy for a fluid membrane that includes
all boundary contributions is much harder to obtain and remains an open problem to which
the variational technique developed in this paper might be successfully applied.
Appendix A. Coordinates and extending stripe
A smooth extension of the αβ interface can be constructed in the following way. At first, we
consider an array of planes where each plane P is perpendicular to the tangent t of the contact
line Lαβσ in the point of intersection, see figure 2. Furthermore, each plane P intersects
the surface Aαβ in the vicinity of Lαβσ and generates a curve Cαβ = P ∩ A αβ which ends
on Lαβσ . Since the planes do not intersect each other in a sufficiently small neighbourhood
U(r0 ) ∈ R3 of a given point r0 ∈ Lαβσ , any point x ∈ Aαβ ∩ U(r0 ) belongs to exactly one
curve Cαβ .
In the next step, each curve Cαβ of intersection will be continued by a small piece of a
circular arc C¯αβ ⊂ P pointing in the direction of the conormal n of the interface Aαβ . The
curvature of the arc C¯αβ shall be identical to the curvature c⊥ in the end point of Cαβ on
Lαβσ . Hence, Cαβ ∪ C¯αβ is a smooth curve in the plane P perpendicular to t as indicated in
figure 2.
Finally, by collecting all curves C¯αβ into a stripe Sαβ we have constructed a regular
extension of the surface Aαβ in a sufficiently small neighbourhood of r0 . In order to show this
proposition we parametrize the contact line by an arclength parameter and the curve Cαβ ∪
C¯αβ by the arclength parameter s. The coordinates s shall satisfy s > 0 on the continuation
C¯αβ , s < 0 on Cαβ , and consequently s = 0 on Lαβσ . Since by definition, the conormal n
and the tangent t of the contact line Lαβσ are orthonormal we can argue by continuity that
the tangent vectors R1 and R2 are linearly independent in a sufficiently small neighbourhood
V(r0 ) ⊂ U(r0 ). Smoothness of the extended surface A¯αβ = Aαβ ∪ Sαβ is guaranteed by the
construction and the fact that n, t and c⊥ are smooth functions of the arclength parameter on Lαβσ .
A general stability criterion for droplets on structured substrates
11569
Appendix B. Variation of surface normal and mean curvature
We will start our calculation with the variation of the surface normal N of the αβ interface.
As N · N = 1 we find
N · δ 1 N = 0.
(B.1)
Variation of the orthogonality relation Ri · N = 0 gives
0 = δ 1 Ri · N + Ri · δ 1 N
= ∂i ψ + Ri · δ 1 N,
(B.2)
where we used (35). Due to (B.2), we can make an ansatz
δ 1 N = a j Rj
(B.3)
for the variation of the surface normal with numerical coefficients a j . These coefficients can
be determined by inserting the ansatz (B.3) into (B.2), which leads to
a j = −g ij ∂i ψ.
(B.4)
We obtain the final result
δ 1 N = −Rj g ij ∂i ψ = −Ri ∂i ψ = −∇ψ,
(B.5)
where ∇ = Ri ∂i is the covariant operator on A¯αβ . As expected, δ 1 N is oriented perpendicular
to the unit surface normal N of the αβ interface.
Using the covariant operator ∇ = Ri ∂i , the mean curvature M can be expressed as
2M = ∇ · N = Ri · ∂i N,
(B.6)
which leads directly to a first variation
2δ 1 M = δ 1 Ri · ∂i N + Ri · ∂i (δ 1 N).
(B.7)
From variation of the orthogonality relation Ri · Rj = δji we conclude δ 1 Ri · Rj = −Ri · δ 1 Rj
and by use of expression (34) we arrive at
j
j
δ 1 Ri · ∂i N = δ 1 Ri · Rj hi = −δ 1 Rj · Ri hi .
(B.8)
The relation R · N = g Rj · N = 0 provides together with the first variation (35) a further
transformation into
i
ij
j
j
j
δ 1 Ri · ∂i N = −Rk · Ri hi hkj ψ − N · Ri hi ∂j ψ = −hi hij ψ.
(B.9)
By explicit calculation one may check that the double sum can be rewritten as
j
−hij hi = 2G − 4M 2 .
(B.10)
Together with (B.5) the second term on the right-hand side of equation (B.7) yields the simple
expression
Ri · ∂i (δ 1 N) = −Ri · ∂i Rj ∂j ψ = −∇ 2 ψ,
(B.11)
with the Laplace–Beltrami operator ∇ 2 = ∇ · ∇ on the A¯αβ . Finally, we arrive at the first
variation
δ 1 M = − 12 ∇ 2 ψ + (G − 2M 2 )ψ
of mean curvature.
(B.12)
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M Brinkmann et al
Appendix C. Variation of conormal and geodesic curvature
To compute the first variation of the conormal m, we make an ansatz
δc1 m = am m + at t + aM Mc ,
(C.1)
with three unknown coefficients am , at and aM . Scalar multiplication of (C.1) with m and
using the variation of m · m = 1 gives the first coefficient
am = m · δc1 m = 0.
(C.2)
By scalar multiplication of equation (C.1) with t and, again, using the variation of an
orthogonality relation t · m = 0 we find
at = t · δc1 m = −m · δc1 t,
(C.3)
where the first variation of the tangent vector
δc1 t = δc1 (ṙ/|ṙ|) = m δ 1 Ṫ + ṁ δ 1 T − t cg∗ δ 1 T
(C.4)
can be calculated using (43). To proceed, we make an ansatz for the derivative of the conormal
ṁ = bm m + bt t + bM Mc
(C.5)
with coefficients bm , bt and bM and find the first and the second coefficient to be
bm = m · ṁ = 0
and
bt = t · ṁ = cg∗ .
(C.6)
Using the orthogonality Mc · m = 0 the third coefficient can be transformed into
bM = Mc · ṁ = −Ṁc · m.
(C.7)
c
The total derivative of the surface normal M of the substrate σ on the contact line with respect
to the arclength is given by
Ṁc = m ω∗ + t c∗ ,
(C.8)
where ω∗ is the torsion along the contact line and c∗ the normal curvature parallel to the
contact line. Thus, we have
bM = −ω∗
(C.9)
for the third coefficient and consequently
ṁ = t cg∗ − Mc ω∗ .
(C.10)
Inserting the relation above into equation (C.4), the first variation of the tangent becomes
δc1 t = mδc1 ṫ − Mc ω∗ δ 1 T .
(C.11)
Consequently, the second coefficient in the ansatz (C.1) is
at = −δ 1 Ṫ .
(C.12)
The third and the last coefficient can be transformed into
aM = δc1 m · Mc = −m · δc1 Mc .
(C.13)
The first variation of the surface normal to the substrate σ under a variation of the contact line
is
∗ 1
δ T,
δc1 Mc = mc⊥
(C.14)
and we finally arrive at
∗ 1
δ T.
aM = −c⊥
(C.15)
A general stability criterion for droplets on structured substrates
11571
Reinserting all coefficients back into ansatz (C.1) gives
∗ 1
δc1 m = −t δ 1 Ṫ − Mc c⊥
δ T
(C.16)
for the first variation of the conormal m to the wetted surface of σ .
Our next task will be to calculate the variation of the geodesic curvature cg∗ with respect
to the surface of σ . By the use of (43) and the definition (44) of the geodesic curvature we
have
δc1 cg∗ = −m · δc1 ṫ − ṫ · δc1 m − cg∗ 2 δ 1 T .
(C.17)
From the orthogonality t · Mc = 0, we can easily derive
ṫ · Mc = −t · Ṁc = −c∗
(C.18)
and end up with
∗ 1
∗ ∗ 1
−ṫ · δc1 m = ṫ · Mc c⊥
δ T = −c⊥
c δ T
(C.19)
for the second term on the right-hand side of equation (C.17). In order to calculate the first
term, we derive the first variation of the tangent vector (C.11) with respect to the arclength
parameter, yielding
δc1 ṫ = ṁ δ 1 Ṫ + m δ 1 T̈ − (m ω∗ 2 + t c∗ ω∗ + Mc ω˙∗ )δ 1 T
and eventually, after scalar multiplication with the conormal m, the relation
−m · δc1 ṫ = ω∗ 2 δ 1 T − δ 1 T̈ .
(C.20)
Using the definition of the Gaussian curvature
∗ ∗
G∗ = c⊥
c − ω∗ 2
on the surface of σ in points of the contact line we finally arrive at
δc1 cg∗ = −δ 1 T̈ − cg∗ 2 + G∗ δ 1 T
(C.21)
(C.22)
for the first variation of the geodesic curvature of the contact line with respect to σ .
Appendix D. Volume constraint for stationary droplet configurations
If we like to enforce the volume constraint up to a second order in ε we have to introduce a
normal variation of the αβ interface
δR = (εψ + ε2 φ)N
(D.1)
with fields ψ and φ such that δ 1 Vβ {ψ} = 0 in a linear order and δ 1 Vβ {φ} + δ 2 Vβ {ψ} = 0 in
quadratic order in the small parameter ε. Inserting the field χ = ψ + εφ into an expansion of
the free energy
F̃ = F̃0 + δ 1 F̃ {χ }ε + 12 δ 2 F̃ {χ }ε 2 + O(ε 3 )
(D.2)
and rearranging the terms in powers of ε shows that only the field ψ appears in the quadratic
order of the expansion (D.2) of the free energy F̃ as long as the preceding linear term
vanishes. Hence, it is sufficient to enforce the volume constraint up to a linear order if we
like to consider the restricted second variation (82) for stationary droplet configurations in the
volume ensemble.
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M Brinkmann et al
Appendix E. Van der Waals contribution to line tension
In this appendix, we will briefly consider the wetting of homogeneous substrate surfaces in
the presence of van der Waals forces. Such interactions can be included in the framework of
effective interface models which include long-ranged interface potentials [1, 25]. In general,
van der Waals forces may favour incomplete or complete wetting corresponding to non-zero
and zero contact angle, θ , respectively. For partial wetting and non-zero contact angle, these
forces are described by an attractive interface potential U (l) which has a minimum at l = lmin
and decays as
AH 1
U (l) ≈
(E.1)
12π l 2
for large l with the Hamaker constant AH [26]. Such an interface potential must also have a
point of inflection which we denote by l = lpin .
The potential depth Umin ≡ U (lmin ) is related to the interfacial free energies via
|Umin | = αβ + βσ − ασ = αβ (1 − cos(θ )) ≈ 12 αβ θ 2
where the asymptotic equality reflects the small gradient expansion.
Within this effective interface model, the line tension is given by
∞
lpin
U (l)
≈ 2
αβ
dl U (l) + |Umin | +
dl √
.
U (l) + |Umin |
lpin
lmin
(E.2)
(E.3)
The first integral represents the contribution from the short-ranged forces as previously
discussed in [1]. The second integral represents the contribution from the long-ranged forces.
If we insert the asymptotic behaviour U (l) ≈ AH /(12π l 2 ) and the small gradient
expression |Umin | ≈ 12 αβ θ 2 into the second integral, we get the line tension contribution
vdW ≈
1 AH
6π θ lpin
(E.4)
from the van der Waals forces. This contribution becomes large in the limit of small contact
angles θ , i.e., in the limit of complete wetting. For θ 1/6π 3◦ , this line tension
contribution satisfies vdW AH / lpin .
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